Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transmission and Reflection coefficients for Schrödinger Operators with Truncated Periodic Potentials that support defect states

Joseph C. Stellman, Jeremy L. Marzuola

Abstract

We consider scattering waves through truncated periodic potentials with perturbations that support localized gap eigenstates. In a small complex neighborhood around an assumed positive bound state of the model operator, we prove the existence of a distinct zero reflection state, or transmission resonance. We compare its location to a previously found scattering resonance and use the properties of solutions near these interesting points to analyze the behavior of transmission and reflection coefficients of scattering solutions near the assumed bound state. By example, we also discuss the truncated simple harmonic oscillator and compare the analysis to the crystalline case.

Transmission and Reflection coefficients for Schrödinger Operators with Truncated Periodic Potentials that support defect states

Abstract

We consider scattering waves through truncated periodic potentials with perturbations that support localized gap eigenstates. In a small complex neighborhood around an assumed positive bound state of the model operator, we prove the existence of a distinct zero reflection state, or transmission resonance. We compare its location to a previously found scattering resonance and use the properties of solutions near these interesting points to analyze the behavior of transmission and reflection coefficients of scattering solutions near the assumed bound state. By example, we also discuss the truncated simple harmonic oscillator and compare the analysis to the crystalline case.
Paper Structure (12 sections, 13 theorems, 145 equations, 5 figures)

This paper contains 12 sections, 13 theorems, 145 equations, 5 figures.

Table of Contents

  1. Introduction
  2. States of Zero Reflection Near Eigenvalues of $H$
  3. Proof of \ref{['thm:zeroreflection']}
  4. Preliminaries for Proofs of \ref{['lem:detbound', 'lem:thetaestimates']}
  5. Proof of \ref{['lem:thetaestimates']}
  6. Proof of \ref{['lem:detbound']}
  7. Proof of \ref{['cor:reflectionasymptotics']}
  8. A Comparison of State Locations
  9. Behavior of Reflection and Transmission Near Eigenvalues of $H$
  10. Case Study: The Truncated Harmonic Oscillator
  11. Proof of \ref{['thm:THOmain']}
  12. Proof of \ref{['lem:locationcomparison']}

Key Result

Theorem 1.3

Let $V(x)$ satisfy assn:regularity, let $\rho>0$ as in eq:compact, and $\Phi$ be a bound state with eigenvalue $E>0$ as in eq:boundstate. Then, for $k>0$ as defined in eq:uedecay-eq:vegrowth and $M_0>\rho\geq0$ such that for all $M\geq M_0$, there exists a unique zero reflection state (or transmissi

Figures (5)

  • Figure 1: Left: Plot of the periodic potential with defect, $V=10+5\cos(4\pi x)+5\tanh(x)\cos(2\pi x)$, truncated at $M=10$. Right: Plot of the associated transmission-reflection (red-blue) curves around the eigenvalue $E\approx 19.77$
  • Figure 1: Left: Plot of the simple harmonic oscillator potential ($V=1+x^2$) truncated at $M=5$. Right: The resulting transmission-reflection curves near the first eigenvalue $E=2$ on the right.
  • Figure 2: A transmission-reflection plot of the same potential in \ref{['fig:periodic']}, for a large set of energies. Outside an interval surrounding the bound state with a sharp transmission peak, we see a band-gap structure from the periodic operator's spectrum.
  • Figure 2: Transmission-reflection plot for the first three eigenvalues of the harmonic oscillator presented in \ref{['fig:harmonic']} truncated at $M=3$. Transmission peak widths increase as $n$ increases.
  • Figure 3: The same harmonic oscillator as in \ref{['fig:harmonic', 'fig:harmonicSweep']}, truncated at $M=3$. As energy increases, behavior settles out to pure transmission. Compare to \ref{['fig:periodicBandGap']}, where the periodic structure leads to bands and gaps in the transmission/reflection plots.

Theorems & Definitions (24)

  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.3
  • Proof 1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 14 more